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arXiv:1006.3343 · Europhysics Letters (EPL)

Adiabatic multicritical quantum quenches: Continuously varying exponents depending on the direction of quenching

Victor Mukherjee, Amit Dutta

Published Nov 1, 2010• 39 citations
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Abstract

We study adiabatic quantum quenches across a quantum multicritical point (MCP) using a quenching scheme that enables the system to hit the MCP along different paths. We show that the power-law scaling of the defect density with the rate of driving depends non-trivially on the path, i.e. , the exponent varies continuously with the parameter α that defines the path, up to a critical value α=α c ; on the other hand for α⩾α c , the scaling exponent saturates to a constant value. We show that dynamically generated and path (α)- dependent effective critical exponents associated with the quasicritical points lying close to the MCP (on the ferromagnetic side), where the energy-gap is minimum, lead to this continuously varying exponent. The scaling relations are established using the integrable transverse XY spin chain and generalized to a MCP associated with a d -dimensional quantum many-body systems (not reducible to two-level systems) using adiabatic perturbation theory. We also calculate the effective path-dependent dimensional shift d 0 (α) (or the shift in center of the impulse region) that appears in the scaling relation for special paths lying entirely in the paramagnetic phase. The numerically obtained results are in good agreement with analytical predictions.

DOI
10.1209/0295-5075/92/37004

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We investigate nonequilibrium dynamical scaling in adiabatic quench processes across quantum multicritical points. Our analysis shows that the resulting power-law scaling depends sensitively on the control path and that anomalous critical exponents may emerge depending on the universality class. We argue that the observed anomalous behavior originates in the fact that the dynamical excitation process takes place asymmetrically with respect to the static multicritical point and that noncritical energy modes may play a dominant role. As a consequence, dynamical scaling requires introducing genuinely nonstatic exponents.
We study the geometric phase of the ground state in a one-dimensional transverse XY spin chain in the vicinity of a quantum multi-critical point. We approach the multi-critical point along different paths and estimate the geometric phase by applying a rotation in all spins about the z axis by an angle η. Although the geometric phase itself vanishes at the multi-critical point, the derivative with respect to the anisotropy parameter of the model shows peaks at different points on the ferromagnetic side close to it where the energy gap is a local minimum; we call these points 'quasi-critical'. The value of the derivative at any quasi-critical point scales with the system size in a power-law fashion with the exponent varying continuously with the parameter α that defines a path, up to a critical value α = α c = 2. For α > α c , or on the paramagnetic side, no such peak is observed. Numerically obtained results are in perfect agreement with analytical predictions.
We study the generation of defects when a quantum spin system is quenched through a multicritical point by changing a parameter of the Hamiltonian as t /τ, where τ is the characteristic timescale of quenching. We argue that when a quantum system is quenched across a multicritical point, the density of defects ( n ) in the final state is not necessarily given by the Kibble–Zurek scaling form n ∼1/τ d ν/( z ν+1) , where d is the spatial dimension, and ν and z are respectively the correlation length and dynamical exponent associated with the quantum critical point. We propose a generalized scaling form of the defect density given by n ∼1/τ d /(2 z 2 ) , where the exponent z 2 determines the behavior of the off-diagonal term of the 2 × 2 Landau–Zener matrix at the multicritical point. This scaling is valid not only at a multicritical point but also at an ordinary critical point.
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When a quantum system exhibiting a second order phase transition is quenched across the critical point in large but finite time, the dynamics are not adiabatic in the critical region and the Kibble-Zurek (KZ) mechanism provides a framework to determine local observables such as the mean defect density. However, to find higher-point functions, one has to go beyond the KZ paradigm asshown in recent works on one-dimensional transverse field Ising model (TFIM) following a linear quench. It has been found that (i) besides the KZ scale, the quench dynamics depend on another length scale that arises due to the finite phase difference between the low energy modes, and (ii) contrary to the expectations based on the KZ mechanism, in general, the correlation functions do not decay exponentially with distance. Motivated by these results for the linear quench, we are interested in understanding if these properties are universal, and consider the 1D TFIM when the transverse field varies algebraically in the vicinity of the critical field. We focus on the equal-time,longitudinal kink-kink correlation function at the end of the quench from the paramagnetic to the ferromagnetic phase, and find that (i) the correlator depends only on the KZ length for superlinear quenches, otherwise an additional dephasing length is required to describe it, and (ii) the dephased correlator decays as a compressed exponential with an exponent that changes continuously with the quench exponent. Our results are obtained using an adiabatic perturbation theory, analytical arguments and exact numerical integration of the relevant equations.